Question

Tangents are drawn from points on line $x - y + 3 = 0$ to a parabola ${y^2} = 8x$. Then the variable chord of contact passes through a fixed point whose coordinates will be?

Answer 1

Hint: We use the method of substitution to find a point on the given line. Use the equation of chord of contact with respect to a given point of a parabola and substitute the values obtained from the point. Solve to get the value of coordinates.

Chord of contact with respect to point $({x_1},{y_1})$ of a parabola ${y^2} = 4ax$ is given

by $y{y_1} = 2a(x + {x_1})$.

Complete step-by-step answer:

We are given the equation of line as $x - y + 3 = 0$

The point through which tangents are drawn to the parabola lies on the line. We use equation of line

to find the point.

Let us assume the value of x-coordinate as a variable, say $x = k$.

Then substitute the value of x in the equation of line to find the y-coordinate.

$\Rightarrow k - y + 3 = 0$

Shift y to RHS of the equation

$\Rightarrow k + 3 = y$

So, the point on the line $x - y + 3 = 0$ is $(k,k + 3)$. …………..… (1)

Now we are given an equation of parabola as ${y^2} = 8x$.

Compare the equation of parabola with general equation of parabola i.e. ${y^2} = 4ax$

$\Rightarrow 4a = 8$

$\Rightarrow a = 2$ ………..… (2)

Since we know that the equation of chord of contact is given by$y{y_1} = 2a(x + {x_1})$.

Substitute the value of point from equation (1) and value of a from equation (2)

Put ${x_1} = k,{y_1} = k + 3,a = 2$

$\Rightarrow y(k + 3) = 2 \times 2(x + k)$

$\Rightarrow yk + 3y = 4x + 4k$

Bring all the terms to LHS of the equation

$\Rightarrow yk + 3y - 4x - 4k = 0$

Collect the terms having value ‘k’ common

$\Rightarrow 3y - 4x + (yk - 4k) = 0$

Take ‘k’ common from second bracket

$\Rightarrow (3y - 4x) + k(y - 4) = 0$

Since ‘k’ is a constant value, therefore if we can take $(3y - 4x) = {P_1};(y - 4) = {P_2}$as two straight lines

$\Rightarrow {P_1} + k{P_2} = 0$

Now the point will pass through the intersection of straight lines ${P_1},{P_2}$

Put ${P_2} = 0$

$\Rightarrow y - 4 = 0$

Shifting the value of constant term to RHS

$\Rightarrow y = 4$

Put ${P_1} = 0$

$\Rightarrow 3y - 4x = 0$

Substitute the value of ‘y’ as 4

$\Rightarrow 3 \times 4 - 4x = 0$

$\Rightarrow 12 - 4x = 0$

Shifting the value of constant term to RHS

$\Rightarrow - 4x = - 12$

Divide both sides by -4

$\Rightarrow \dfrac{{ - 4x}}{{ - 4}} = \dfrac{{ - 12}}{{ - 4}}$

Cancel same terms from numerator and denominator on both sides of the equation

$\Rightarrow x = 3$

$\therefore$The point becomes $(3,4)$

Note: Chord joining the two points of contact from the tangents to the parabola from same external points is called the chord of contact. It is of the form$T = 0$. PQ is the chord of contact.

Students might make mistake in the solution as they take the value of ‘a’ as 8 when they see

the equation of parabola, which is wrong. Keep in mind we compare the equation with the general equation and then find the value of ‘a’.