The sum of diameters of the circles that touch (i) the parabola 75x2 = 64(5y – 3) at the point (\(\frac{8}{5},\frac{6}{5}\)) and (ii) the y-axis is equal to _______.
Correct Answer: 10
Solution and Explanation
Equation of tangent to the parabola at
P(\(\frac{8}{5},\frac{6}{5}\))
75x⋅85=160(y+\(\frac{6}{5}\))−192
⇒ 120x = 160y
⇒ 3x = 4y
The equation of the circle touching the given parabola at P can be taken as
(x−\(\frac{8}{5}\))2+(y−\(\frac{6}{5}\))2+λ(3x−4y)=0
If this circle touches the y-axis then
\(\frac{64}{25}\)+(y−65)2+λ(−4y)=0
⇒y2−2y(2λ+65)+4=0
⇒ D = 0
⇒(\(\frac{2λ+6}{6}\))2=4
⇒λ=\(\frac{2}{5}\) or −\(\frac{8}{5}\)
Radius = 1 or 4
Sum of diameter = 10
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Concepts Used:
Parabola
Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).
Standard Equation of a Parabola
For horizontal parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A,
- Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
- P(x,y) as the moving point.
- Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
- The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
- The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
- By definition PM = PS
=> MP2 = PS2
- So, (a + x)2 = (x - a)2 + y2.
- Hence, we can get the equation of horizontal parabola as y2 = 4ax.
For vertical parabola
- Let us consider
- Origin (0,0) as the parabola's vertex A
- Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
- P(x,y) as any moving point
- Let us now draw a perpendicular SZ from S to the directrix.
- Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
- The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
- By definition PM = PS
=> MP2 = PS2
So, (b + y)2 = (y - b)2 + x2
- As a result, the vertical parabola equation is x2= 4by.