Question:
The point on the parabola $y^2 = 64x$ which is nearest to the line $4x + 3y + 35 = 0$ has coordinates
The point on the parabola $y^2 = 64x$ which is nearest to the line $4x + 3y + 35 = 0$ has coordinates
Updated On: Apr 26, 2024
- (9, -24)
- (1, 81)
- (4, -16)
- (-9, -24)
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Given equation of parabola is
$y^{2}=64 x\,\,\,...(i)$
The point at which the tangent to the curve is parallel to the line is the nearest point on the curve.
On differentiating both sides of E (i), we get
$2 y \frac{d y}{d x} =64$
$\Rightarrow \frac{d y}{d x} =\frac{32}{y}$
Also, slope of the given line is $-\frac{4}{3}$.
$\therefore -\frac{4}{3}=\frac{32}{y} $
$\Rightarrow y=-24$
From E (i), $(-24)^{2}=64 x$
$ \Rightarrow x=9$
$\therefore$ Hence, the required point is $(9,-24) .$
$y^{2}=64 x\,\,\,...(i)$
The point at which the tangent to the curve is parallel to the line is the nearest point on the curve.
On differentiating both sides of E (i), we get
$2 y \frac{d y}{d x} =64$
$\Rightarrow \frac{d y}{d x} =\frac{32}{y}$
Also, slope of the given line is $-\frac{4}{3}$.
$\therefore -\frac{4}{3}=\frac{32}{y} $
$\Rightarrow y=-24$
From E (i), $(-24)^{2}=64 x$
$ \Rightarrow x=9$
$\therefore$ Hence, the required point is $(9,-24) .$
Was this answer helpful?
0
0
Top Questions on Application of derivatives
- Evaluate the integral: $\int_{-\pi}^{\pi} x^2 \sin(x) \, dx$
- VITEEE - 2024
- Mathematics
- Application of derivatives
- For the function $f(x) = x^3 - 6x^2 + 12x - 3$, $x = 2$ is:}
- KCET - 2024
- Mathematics
- Application of derivatives
- If \( \frac{dy}{dx} - y \log_e 2 = 2^{\sin x} (\cos x - 1) \log_e 2 \), then \( y \) is:
- BITSAT - 2024
- Mathematics
- Application of derivatives
- The solution of the differential equation \( (x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^2 \) is:
- BITSAT - 2024
- Mathematics
- Application of derivatives
- If the area bounded by the curves \( y = ax^2 \) and \( x = ay^2 \) (where \( a>0 \)) is 3 sq. units, then the value of \( a \) is:
- BITSAT - 2024
- Mathematics
- Application of derivatives
View More Questions
Questions Asked in WBJEE exam
- If 1000! = 3n × m, where m is an integer not divisible by 3, then n =?
- WBJEE - 2024
- Probability
- Chords AB CD of a circle intersect at right angle at the point P. If the lengths of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is:
- If the relation between the direction ratios of two lines in \(\mathbb{R}^3\) are given by \(l + m + n = 0\), \(2lm + 2mn - ln = 0\), then the angle between the lines is:
- WBJEE - 2024
- Vectors
- If \(a, b, c\) are distinct odd natural numbers, then the number of rational roots of the equation \(ax^2 + bx + c = 0\) is:
- WBJEE - 2024
- Quadratic Equation
- What force \(F\) is required to start moving this 10 kg block shown in the figure if it acts at an angle of \(60^\circ\) as shown? (\(\mu_s = 0.6\)).
- WBJEE - 2024
- laws of motion
View More Questions
Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
Read More: Application of Derivatives