𝑃𝑃(𝑎𝑎𝑝𝑝 2, 2𝑎𝑎𝑝𝑝) and 𝑄𝑄(𝑎𝑎𝑞𝑞 2, 2𝑎𝑎𝑞𝑞) are 2 points on the parabola 𝑦𝑦 2 = 4𝑎𝑎𝑥𝑥.a) Show that the equation of the chord 𝑃𝑃𝑄𝑄 is 2𝑥𝑥 − (𝑝𝑝 + 𝑞𝑞)𝑦𝑦 + 2𝑎𝑎𝑝𝑝𝑞𝑞 = 0.b) If 𝑂𝑂𝑃𝑃 ⊥ 𝑂𝑂𝑄𝑄, show that the 𝑥𝑥-intercept of 𝑃𝑃𝑄𝑄 is independent of 𝑝𝑝 and 𝑞𝑞.chord

a) The coordinates of points P and Q are given by (ap^2, 2ap) and (aq^2, 2aq) respectively. The equation of the line passing through these two points can be found using the formula for a line in 2D space:

y - y1 = m(x - x1)

where m is the slope of the line, (x1, y1) are the coordinates of a point on the line. The slope m is given by (y2 - y1) / (x2 - x1).

Substituting the given points into the formula, we get:

y - 2ap = [(2aq - 2ap) / (aq^2 - ap^2)] * (x - a*p^2)

Simplifying this equation gives us the equation of the chord PQ:

2x - (p + q)y + 2ap*q = 0

b) If OP is perpendicular to OQ, then the product of their slopes is -1. The slope of OP is 2p/ap^2 = 2/ap and the slope of OQ is 2q/aq^2 = 2/aq. Therefore, (2/ap) * (2/aq) = -1.

Solving for pq gives us pq = -a^2/4. Substituting this into the equation of the chord PQ from part a) gives us the x-intercept (when y = 0):

2x - 2a*(-a^2/4) = 0

Solving for x gives us x = a^2/4, which is independent of p and q.

a) The coordinates of points P and Q are given by (ap^2, 2ap) and (aq^2, 2aq) respectively. The equation of the line passing through these two points can be found using the formula for a line passing through two points (x1, y1) and (x2, y2):

(y - y1) = [(y2 - y1) / (x2 - x1)] * (x - x1)

Substituting the coordinates of P and Q into this formula gives:

(y - 2ap) = [(2aq - 2ap) / (aq^2 - ap^2)] * (x - a*p^2)

Simplifying this equation gives:

2x - (p + q)y + 2ap*q = 0

which is the equation of the chord PQ.

b) If OP is perpendicular to OQ, then the product of their slopes is -1. The slope of OP is 2ap / ap^2 = 2/p and the slope of OQ is 2aq / aq^2 = 2/q. Therefore, 2/p * 2/q = -1, which simplifies to p*q = -1.

The x-intercept of the line is the value of x when y = 0. Substituting y = 0 into the equation of the chord gives:

2x - (p + q)0 + 2ap*q = 0

which simplifies to x = -apq. Since p*q = -1, the x-intercept is a, which is independent of p and q.