If the line y = 4 + kx, k > 0, is the tangent to the parabola y = x – x2 at the point P and V is the vertex of the parabola, then the slope of the line through P and V is:
- \(\frac{3}{2}\)
- \(\frac{26}{9}\)
- \(\frac{5}{2}\)
- \(\frac{23}{6}\)
The Correct Option is A
Solution and Explanation
The correct option is(C): \(\frac{5}{2}\).
∵ Line y = kx + 4 touches the parabola y = x – x2.
So, kx + 4 = x – x2 ⇒ x2 + (k – 1) x + 4 = 0 has only one root
(k – 1)2 = 16 ⇒ k = 5 or – 3 but k > 0
So, k = 5.
And hence x2 + 4x + 4 = 0 ⇒ x = – 2
So, P(–2, –6) and V is
\((\frac{1}{2},\frac{1}{4})\)
Slope of PV
\(=\frac{\frac{1}{4}+6}{\frac{1}{2}+2}=\frac{5}{2}\)
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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.